NACA 0012 AIRFOIL



 

    The NACA 0012 airfoil has been designed for supersonic flows, so we used the AVBP code to simulate the flow over an airfoil NACA 0012 with an incoming supersonic flow ( Mach=1.2 ). We tested and compatared 3 kinds of meshes: a coarse structured mesh, a fine structured mesh, and a unstructured triangular mesh. We used a triangular mesh to simulate the flow over the wing profile for different angles of attack: we visualized the velocity and pressure field for the different cases. Even if in this case we worked with the euler equations, we used our simulations to evaluate the polar curve of the wing ( lift coefficient Cp versus alpha, with alpha the angle of attack ): the Cp coefficient is obtained by integrating the surface pressure coefficient distribution.
 

COMPARISON FOR DIFFERENT MESHES

    In order to compare the 3 kinds of meshes, we simulated the same flow (angle of attack 7 degrees) with the 3 meshes.

Coarse structured mesh

    The coarse structured mesh shown below allows a quick convergence.

The coarse mesh.

Close-up of the coarse mesh.
The mesh is finer close to the airfoil, so that gradient phenomena can be captured.
The simulation with the coarse mesh give the following velocity and pressure fields:
 
 


 

Mach field with the coarse mesh, incidence 7 degrees.
 
 


 

Pressure field with the coarse mesh, incidence 7 degrees.
 
 

Convergence for the coarse mesh.





After 500 iterations, we have a convergence in 10^-3, which is not enough, but we see that we could improve it with more iterations.

 
Fine structured mesh

With more than double of cells used for the coarse mesh, the fine mesh is refined close to the airfoil in order to get more reliable results. However, in terms of CPU units, the calculation time is higher before having a converged solution.

The fine mesh.

Close-up of the fine mesh.


 
 
The simulation with the coarse mesh give the following velocity and pressure fields:
 
 

Mach field with the fine mesh, incidence 7 degrees.
 
 


 

Velocity vectors for the fine mesh, incidence 7 degrees.
 

Pressure field with the fine mesh, incidence 7 degrees.
 
 
 
 
 

Convergence for the coarse mesh




After 1000 iterations, we have a convergence of 10^-4. At 500 iterations, convergence is not better than for the coarse mesh: as the convergence curve is not flat, we could improve it by running more iterations. Of course the fine mesh allows to capture better physical phenomenoma, especially shock waves.
 

Triangular mesh

The triangular mesh.
 

Close-up of the triangular mesh.
 

Mach field with the triangular mesh, incidence 7 degrees.
 
 

Pressure field with the triangular mesh, incidence 7 degrees.
 
 

Convergence for the triangular mesh.

 
After 1000 iterations we have a convergence in 10^-4, which is similar to the fine mesh. The triangular mesh fits better to the wing than the structured mesh, therefore physical phenomena close to the wing are better captured.

We can compare maximum and minimum pressure and velocity for the 3 meshes:
 

 
 Structured coarse mesh
Structured fine mesh
Triangular mesh
Maximum velocity
1.807
1.807
1.8290
Minimum velocity
0.011
0.047
0.019
Maximum pressure
1.1988
1.1966
1.1949
Minimum pressure
0.2118
0.2091
0.2090
The 3 meshes give similar results: they are different only at the second decimal.